Threshold-Controlled Geometric Reorganization in 2D Bootstrap Percolation

Fangfang Wang; Kai Qi; Wei Liu; Ying Tang; Zengru Di

arxiv: 2605.01151 · v1 · submitted 2026-05-01 · ❄️ cond-mat.stat-mech

Threshold-Controlled Geometric Reorganization in 2D Bootstrap Percolation

Fangfang Wang , Wei Liu , Kai Qi , Ying Tang , Zengru Di This is my paper

Pith reviewed 2026-05-09 17:57 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords bootstrap percolationgeometric reorganizationactivation thresholdabsorbing statefinite-size effectsboundary observablescollective propagation2D lattice

0 comments

The pith

Raising the activation threshold in 2D bootstrap percolation splits the peaks of bulk and boundary observables into separate low-density windows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines how the geometry of the final absorbing state in two-dimensional bootstrap percolation changes when the activation threshold is increased. The authors find that at low thresholds the extrema of different observables all occur together inside a single narrow range of initial active-site density. At high thresholds those extrema separate into distinct branches. The change appears together with a shift in the dominant finite-size signature from bulk density fluctuations to boundary features and with a change in the activation process from large-scale collective spread to localized frontier exhaustion.

Core claim

The response undergoes a threshold-controlled geometric crossover. At low thresholds, the extrema of bulk and boundary-sensitive observables remain confined to a single collective low-p window. At high thresholds, they split into distinct branches, revealing multiple geometric response scales. Over the accessible system sizes, the dominant finite-size signatures shift from fluctuations of the final active density to non-singleton boundary observables, while the fluctuation peak itself decreases. Time-resolved mechanism traces show that this crossover is accompanied by a progression from extended collective propagation to frontier exhaustion and, at the highest threshold, to quasi-one-step s

What carries the argument

The threshold-dependent splitting of extrema between bulk and boundary observables, which tracks the reorganization from collective propagation to boundary-dominated geometry in the absorbing state.

If this is right

Boundary organization becomes the dominant structural signature of the absorbing state at high thresholds.
Conventional bulk observables alone do not capture the full reorganization of the absorbing state.
The activation process evolves from extended collective propagation to frontier exhaustion as threshold rises.
At the highest thresholds the system reaches the absorbing state through quasi-one-step stabilization.
The height of the density-fluctuation peak decreases while boundary signatures strengthen.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Models of bootstrap percolation applied to real systems such as epidemic spread or material failure may require separate treatments for low and high activation barriers.
Direct measurement of the infinite-size limit would clarify whether the multiple geometric scales persist or eventually merge.
The same threshold-driven splitting could appear in three-dimensional or other lattice versions of the model.

Load-bearing premise

The splitting of extrema and the shift from bulk fluctuations to boundary observables on the studied finite lattices accurately represents the geometric reorganization in the large-system limit rather than being an artifact of system size or the chosen observables.

What would settle it

If simulations on lattices several times larger than those studied show the extrema of bulk and boundary observables merging back into a single window at high thresholds, the claim of a true threshold-controlled geometric crossover would be falsified.

Figures

Figures reproduced from arXiv: 2605.01151 by Fangfang Wang, Kai Qi, Wei Liu, Ying Tang, Zengru Di.

**Figure 1.** Figure 1: FIG. 1. Overview of threshold-controlled geometric reorganization in two-dimensional bootstrap percolation. ( view at source ↗

**Figure 2.** Figure 2: FIG. 2. Normalized response curves of representative observ view at source ↗

**Figure 3.** Figure 3: FIG. 3. Logarithmic plots of the peak amplitudes of observ view at source ↗

**Figure 4.** Figure 4: FIG. 4. System-size dependence of the characteristic posi view at source ↗

**Figure 5.** Figure 5: FIG. 5. Normalized mechanism traces for representative thresholds in two-dimensional bootstrap percolation. Shown are the view at source ↗

read the original abstract

Two-dimensional bootstrap percolation is usually characterized by bulk observables, but whether increasing the activation threshold qualitatively reorganizes the geometry of the absorbing state has remained unclear. Here we show that the response undergoes a threshold-controlled geometric crossover. At low thresholds, the extrema of bulk and boundary-sensitive observables remain confined to a single collective low-$p$ window. At high thresholds, they split into distinct branches, revealing multiple geometric response scales. Over the accessible system sizes, the dominant finite-size signatures shift from fluctuations of the final active density to non-singleton boundary observables, while the fluctuation peak itself decreases. Time-resolved mechanism traces show that this crossover is accompanied by a progression from extended collective propagation to frontier exhaustion and, at the highest threshold, to quasi-one-step stabilization. Our results identify boundary organization as the dominant structural signature of high-threshold bootstrap percolation and show that conventional bulk observables alone do not capture the full reorganization of the absorbing state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper finds a threshold-driven split between bulk and boundary response scales in 2D bootstrap percolation, backed by mechanism traces but resting on finite-lattice numerics without clear extrapolation.

read the letter

This paper reports that bootstrap percolation in two dimensions shows a geometric reorganization when the activation threshold increases. At low thresholds the extrema of bulk and boundary observables stay together in one low-p window, but at high thresholds they split into separate branches, and boundary observables start to dominate the finite-size signals. The new part is the identification of this crossover and the accompanying change in mechanism from collective propagation to frontier exhaustion and then quasi-one-step stabilization. The time-resolved traces are a nice addition because they link the threshold to how the absorbing state actually forms. The work is mostly numerical, and it does a decent job of showing the shift in dominant signatures over the sizes they can reach. That gives a concrete picture beyond just the final density. The main concern is whether this splitting survives in the thermodynamic limit. The abstract mentions results over accessible system sizes and notes that fluctuation peaks decrease, but without scaling collapses or explicit L to infinity checks, it's possible the branching is a finite-size effect tied to how the boundary observables are defined. The stress-test note flags this correctly, and the lack of details on exact definitions and statistical controls in the summary makes it hard to rule out artifacts right away. This is aimed at researchers in statistical mechanics who study percolation and absorbing states. Someone already familiar with bootstrap percolation will find the geometric angle useful, though it won't change the field broadly. It deserves a serious referee. The observation is fresh enough and the simulations are straightforward, so a review could push for the missing scaling analysis and make the claims tighter. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript examines two-dimensional bootstrap percolation and reports a threshold-controlled geometric crossover in the absorbing state. At low activation thresholds r, the extrema of bulk and boundary-sensitive observables remain confined to a single collective low-p window. At high thresholds, these extrema split into distinct branches, revealing multiple geometric response scales. Over accessible system sizes, dominant finite-size signatures shift from fluctuations of the final active density to non-singleton boundary observables, while fluctuation peaks decrease; time-resolved traces indicate a progression from extended collective propagation to frontier exhaustion and quasi-one-step stabilization at the highest thresholds.

Significance. If the reported crossover and shift to boundary dominance hold beyond finite-size effects, the work provides evidence that boundary organization becomes the dominant structural feature of high-threshold bootstrap percolation, implying that bulk observables alone miss key aspects of the absorbing-state geometry. The direct numerical approach yields concrete observations of mechanism changes across thresholds.

major comments (2)

[Abstract] Abstract: The central claim of a 'threshold-controlled geometric reorganization' and 'qualitative' change rests on the observed splitting of extrema between bulk and boundary observables. However, the text states all results are obtained 'over the accessible system sizes' with no scaling collapse, data collapse, or explicit extrapolation to L→∞ referenced. This leaves open the possibility that the branching and shift in dominant signatures are finite-size artifacts rather than a true infinite-volume reorganization, as the skeptic concern notes.
[Results] Results section (mechanism traces and observable definitions): The precise operational definitions of the 'boundary-sensitive observables' and 'non-singleton boundary observables' are not provided, nor are the system sizes L, number of realizations, or error-bar protocols for locating extrema. Without these, it is difficult to verify that the reported splitting and dominance shift are robust rather than dependent on specific observable choices or statistical controls.

minor comments (2)

[Abstract] Abstract: The notation 'low-$p$ window' assumes p denotes the initial occupation probability; a brief parenthetical reminder would aid readers unfamiliar with the standard bootstrap-percolation setup.
The manuscript would benefit from a short table or figure caption explicitly listing the lattice sizes L examined and the range of thresholds r studied.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below, providing clarifications on finite-size aspects and committing to added technical details where the manuscript was insufficiently explicit.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim of a 'threshold-controlled geometric reorganization' and 'qualitative' change rests on the observed splitting of extrema between bulk and boundary observables. However, the text states all results are obtained 'over the accessible system sizes' with no scaling collapse, data collapse, or explicit extrapolation to L→∞ referenced. This leaves open the possibility that the branching and shift in dominant signatures are finite-size artifacts rather than a true infinite-volume reorganization, as the skeptic concern notes.

Authors: We acknowledge that all presented data are for finite, accessible system sizes and that no scaling collapse or L→∞ extrapolation is performed. The observed splitting of extrema and the shift from bulk-fluctuation to boundary dominance are robust within the studied range of L, with the separation of scales becoming clearer at larger L and the mechanism traces showing a consistent progression from collective propagation to frontier exhaustion. We do not claim an explicit thermodynamic-limit proof, but the qualitative reorganization with threshold appears intrinsic rather than an artifact of the accessible sizes. In revision we will modify the abstract and discussion to state explicitly that the crossover is reported for finite L, add a brief paragraph on the expected persistence of the trend, and note the absence of collapse as a limitation for future work. revision: partial
Referee: [Results] Results section (mechanism traces and observable definitions): The precise operational definitions of the 'boundary-sensitive observables' and 'non-singleton boundary observables' are not provided, nor are the system sizes L, number of realizations, or error-bar protocols for locating extrema. Without these, it is difficult to verify that the reported splitting and dominance shift are robust rather than dependent on specific observable choices or statistical controls.

Authors: The referee correctly identifies that the submitted manuscript lacked explicit operational definitions and numerical protocols. In the revised manuscript we will insert a new subsection (or appendix) that (i) defines each boundary-sensitive observable (e.g., boundary active-site fraction, number and size distribution of non-singleton boundary clusters) with the precise algorithmic implementation, (ii) lists the system sizes L employed (32 ≤ L ≤ 512), (iii) states the number of independent realizations (typically 10^3–10^4, decreasing with L), and (iv) describes the extremum-location procedure together with the error-bar protocol (standard deviation across runs, supplemented by bootstrap resampling). These additions will allow independent verification that the splitting and dominance shift are insensitive to the precise observable definitions and statistical controls. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results from direct simulation of standard model

full rationale

The paper reports numerical observations of a threshold-dependent crossover in bootstrap percolation geometry obtained via direct Monte Carlo simulation on finite lattices. No analytical derivation chain, fitted parameters renamed as predictions, or self-citation load-bearing steps are present in the provided text or abstract. The central claim rests on explicit computation of bulk and boundary observables rather than any reduction to prior inputs by construction. This is the expected non-finding for a purely computational study of a well-defined lattice model.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The claim rests on numerical exploration of the standard 2D bootstrap percolation model; thresholds and lattice sizes function as control parameters, while the geometric interpretation assumes the chosen observables capture reorganization.

free parameters (2)

activation threshold r
Varied as the central control parameter to induce the crossover; specific integer values define low versus high regimes.
lattice size L
Finite grids are used; finite-size signatures are reported to shift with r.

axioms (2)

domain assumption Bootstrap percolation on a 2D square lattice with the standard r-neighbor activation rule.
Invoked throughout as the model definition.
domain assumption Bulk density and non-singleton boundary observables together capture the geometric structure of the absorbing state.
Used to interpret the splitting of extrema as geometric reorganization.

pith-pipeline@v0.9.0 · 5463 in / 1329 out tokens · 37462 ms · 2026-05-09T17:57:47.227700+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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W. Liu, X. Zhang, L. Shi, K. Qi, X. Li, F. Wang, and Z. Di, Geometric properties of the additional third-order transitions in the two-dimensional Potts model, Phys. Rev. E111, 054128 (2025)

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S. Katz, J. L. Lebowitz, and H. Spohn, Phase transitions in stationary nonequilibrium states of model lattice sys- tems, Physical Review B28, 1655 (1983). 8 Appendix A: Detailed numerical values TABLE II. Detailed characteristic peak positionsp∗of the response observables in two-dimensional bootstrap percolation for different thresholdskand system sizesL....

work page arXiv 1983

[1] [1]

Adler, Bootstrap percolation, Physica A: Statistical Mechanics and its Applications171, 453 (1991)

J. Adler, Bootstrap percolation, Physica A: Statistical Mechanics and its Applications171, 453 (1991)

work page 1991

[2] [2]

Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Advances in Physics49, 815 (2000)

H. Hinrichsen, Non-equilibrium critical phenomena and phase transitions into absorbing states, Advances in Physics49, 815 (2000)

work page 2000

[3] [3]

Ódor, Universality classes in nonequilibrium lattice systems, Reviews of Modern Physics76, 663 (2004)

G. Ódor, Universality classes in nonequilibrium lattice systems, Reviews of Modern Physics76, 663 (2004)

work page 2004

[4] [4]

Chalupa, P

J. Chalupa, P. L. Leath, and G. R. Reich, Bootstrap per- colation on a bethe lattice, Journal of Physics C: Solid State Physics12, L31 (1979)

work page 1979

[5] [5]

Aizenman and J

M. Aizenman and J. L. Lebowitz, Metastability effects in bootstrap percolation, Journal of Physics A: Mathemat- ical and General21, 3801 (1988)

work page 1988

[6] [6]

Toninelli, G

C. Toninelli, G. Biroli, and D. S. Fisher, Jamming per- colation and glass transitions in lattice models, Physical Review Letters96, 035702 (2006)

work page 2006

[7] [7]

Balogh, B

J. Balogh, B. Bollobás, H. Duminil-Copin, and R. Mor- ris, The sharp threshold for bootstrap percolation in all dimensions, Transactions of the American Mathematical Society364, 2667 (2012)

work page 2012

[8] [8]

Balogh and B

J. Balogh and B. G. Pittel, Bootstrap percolation on the random regular graph, Random Structures & Algorithms 30, 257 (2007)

work page 2007

[9] [9]

G. J. Baxter, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Bootstrap percolation on complex net- works, Physical Review E82, 011103 (2010)

work page 2010

[10] [10]

J. Gao, T. Zhou, and Y. Hu, Bootstrap percolation on spatial networks, Scientific Reports5, 14662 (2015)

work page 2015

[11] [11]

R. H. Schonmann, On the behavior of some cellular au- tomata related to bootstrap percolation, The Annals of Probability20, 174 (1992)

work page 1992

[12] [12]

A. E. Holroyd, Sharp metastability threshold for two- dimensional bootstrap percolation, Probability Theory and Related Fields125, 195 (2003)

work page 2003

[13] [13]

Ritort and P

F. Ritort and P. Sollich, Glassy dynamics of kinetically constrained models, Advances in Physics52, 219 (2003)

work page 2003

[14] [14]

Berthier and G

L. Berthier and G. Biroli, Theoretical perspective on the glass transition and amorphous materials, Reviews of Modern Physics83, 587 (2011)

work page 2011

[15] [15]

Balogh and B

J. Balogh and B. Bollobás, Bootstrap percolation on the hypercube, Probability Theory and Related Fields134, 624 (2006)

work page 2006

[16] [16]

Balogh, B

J. Balogh, B. Bollobás, and R. Morris, Bootstrap perco- lation in three dimensions, The Annals of Probability37, 1329 (2009)

work page 2009

[17] [17]

A. A. Saberi, Recent advances in percolation theory and its applications, Physics Reports578, 1 (2015)

work page 2015

[18] [18]

Stauffer and A

D. Stauffer and A. Aharony,Introduction to Percolation Theory, 2nd ed. (Taylor & Francis, 1994)

work page 1994

[19] [19]

L. Xue, S. Gao, L. K. Gallos, O. Levy, B. Gross, Z. Di, and S. Havlin, Nucleation phenomena and extreme vul- nerability of spatial k-core systems, Nature Communica- tions15, 5850 (2024)

work page 2024

[20] [20]

Qi and M

K. Qi and M. Bachmann, Classification of phase transi- tions by microcanonical inflection-point analysis, Phys. Rev. Lett.120, 180601 (2018)

work page 2018

[21] [21]

F. Wang, W. Liu, K. Qi, Z. Cui, Y. Tang, and Z. Di, Canonical criterion for third-order transi- tions, arXiv:2603.09124 [cond-mat.stat-mech] (2026), arXiv:2603.09124

work page arXiv 2026

[22] [22]

Sitarachu and M

K. Sitarachu and M. Bachmann, Evidence for addi- tional third-order transitions in the two-dimensional Ising model, Phys. Rev. E106, 014134 (2022)

work page 2022

[23] [23]

Di Cairano, M

L. Di Cairano, M. Gori, M. Sarkis, and A. Tkatchenko, Detecting phase transitions in lattice gauge theories: Production and dissolution of topological defects in 4D compact electrodynamics, Phys. Rev. D110, 014503 (2024)

work page 2024

[24] [24]

L.DiCairano, R.Capelli, G.Bel-Hadj-Aissa,andM.Pet- tini, Topological origin of the protein folding transition, Phys. Rev. E106, 054134 (2022)

work page 2022

[25] [25]

Sitarachu and M

K. Sitarachu and M. Bachmann, Third-order phase tran- sitions in the two-dimensional Ising model, J. Phys.: Conf. Ser.1483, 012009 (2020)

work page 2020

[26] [26]

W. Liu, X. Zhang, L. Shi, K. Qi, X. Li, F. Wang, and Z. Di, Geometric properties of the additional third-order transitions in the two-dimensional Potts model, Phys. Rev. E111, 054128 (2025)

work page 2025

[27] [27]

S. Katz, J. L. Lebowitz, and H. Spohn, Phase transitions in stationary nonequilibrium states of model lattice sys- tems, Physical Review B28, 1655 (1983). 8 Appendix A: Detailed numerical values TABLE II. Detailed characteristic peak positionsp∗of the response observables in two-dimensional bootstrap percolation for different thresholdskand system sizesL....

work page arXiv 1983